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Games Numbers Play

Fernando Gouvêa broke with convention for the lecture he presented on May 21. Mathematicians giving public lectures know, the Colby College professor explained, that the “person in the street” thinks math is all and only about numbers. To counteract this misconception and to raise awareness of the diversity within mathematics, presenters tend to steer clear of the topic.

But not Gouvêa. He devoted his historical talk, titled “Games Numbers Play,” to showing his audience that, given an inquisitive mind and perhaps a handful or two of pebbles, you can become an active observer of “the strange games that numbers play among themselves.” Doing so, Gouvêa hinted, allows you to transform mathematics into not only a source of amusement but also an endless pursuit.

“Every time you answer a question,” he said, “it tends to throw up a new question that you can then go and play with on the basis of the first one.”

There’s an unwritten rule, Gouvêa noted, “that any historical talk about mathematics has to start with the Greeks,” and, in this regard, he played by the book.

Gouvêa told of how the Greeks’ fascination with numbers—their discovery that the human ear is attuned to ratios between lengths of simultaneously plucked strings and their suspicion that, in a world of flux, numbers might boast a comforting constancy—infected classical scholar H. D. F. Kitto, author of The Greeks, a 1952 book about Greek civilization and culture. In an attempt to appreciate the Grecian frame of mind, Kitto found himself embarking on "an insommia-beguiling excursion," as Kitto put it, into questions of number theory, thereby earning himself “an impressive peep into a new and perfect universe,” Gouvêa said.

He described the pattern that so diverted Kitto:

Start with the product 10 x 10 = 100. Add 1 to the first number, and subtract 1 from the second, then compute the product: 11 x 9 = 99. Repeat: 12 x 8 = 96, and so on. The differences between successive products are the odd numbers 1, 3, 5, 7, and so on.

“Who invited the odd numbers to this game?” Gouvêa asked.

“Somehow the squares are crashing our party as well!” he exclaimed, once the action had unfolded a few more steps.

Using a document camera trained on arrays of glass beads, Gouvêa investigated problems as Plato and his ilk did: visually and tactilely, with nary an equals sign or variable in sight. And with this time-honored, but these days often downplayed, means of mathematical exploration came some algebra bashing. (All in good fun: MAA is due to publish Gouvêa’s A Guide to Algebra this fall. The prolific expositor of mathematics clearly bears the subject no ill will.)

“This is what it looks like algebraically,” Gouvêa said as he put up a slide of a formula that represented symbolically a relationship between squares he’d just exemplified with beads. “It just becomes much more boring.”

As evidence of the perhaps unsuspected power of his math-by-manipulative methods, Gouvêa showed using his beads that no two square arrays of equal dimension can be combined to form a larger square. (Try it.)

“We’ve proved that something is impossible,” he said, “which is sort of a surprising thing to do by thinking about pebbles.”

Indeed, but then “Pebble Land” is a surprising place. As his talk progressed, Gouvêa had his audience defining a prime number in terms of pebbles (“You can’t make a rectangle except in a stupid way”), mulling over the characterization of a lone pebble as a “potential triangle,” and contemplating the existence of deceptively impossible-sounding squares that are also triangles.

Late in the game Gouvêa unveiled the “amazing theorem” that “you can make any number whatsoever by taking up to three triangles, up to four squares, up to five pentagons, up to six hexagons, up to seven heptagons, and so on.”

“That’s sort of kooky,” Gouvêa said before leaving it to listeners to puzzle over the mathematical marvel themselves.

Questions like the ones he’d posed, Gouvêa indicated, can easily lead to problems infamous for their difficulty. Start speculating about the possibility of combining two cubes made out of sugar cubes into one bigger sugar cube, after all, and you’re pondering a piece of Fermat’s last theorem.

The allusion to this problem, which mathematicians took 350 years to solve, provided a tidy segue to Gouvêa’s take-home message: “There are immensely many of these games that numbers play with each other,” he said. “Some of them will be easy; some of them will be tremendously hard.”

Gouvêa concluded: “I want to give you an idea how much fun it can be to find out.”

(Readers keen to find out for themselves are encouraged to watch Gouvêa’s lecture in its entirety (On MAA's YouTube Channel). If possible, have pebbles on hand and play along.) —Katharine Merow

Fernando Q. Gouvêa is the Carter Professor of Mathematics at Colby College. He received his master’s degree from the Universidade de São Paulo in 1981 and his Ph.D. from Harvard in 1987. His doctoral thesis in number theory ended up as the book Arithmetic of p-adic Modular Forms. Gouvêa’s research interests include number theory and arithmetic geometry, with a special focus on modular forms and Galois representations, and the history of mathematics, especially the history of algebra and number theory.